# Scientific Notation and Significant Figures

Notes on scientific notation and significant figures prepared by Dr. Masingale, Le Moyne College Department of Chemistry.

## Scientific Notation

All numbers, regardless of magnitude, can be expressed in the form:
N x 10n
where
• N is a number, either an integer or decimal, between 1 and 10.
• n is a positive or negative integer.
When written in standard form there must be one digit, and only one digit to the left of the decimal point in the number N.
• standard: 1.23x106
• non-standard: 123x104

### Positive exponents

36,600
1. a number greater than 1
exponent of 10 is a positive whole number
2. value of the exponent
number of places the decimal point must be moved so that the notation is in standard form
3. 36,600 x 100
For each place the decimal point is moved to the left, add 1 to the original exponent
3.66 x 104

### Negative exponents

0.00563
1. a number less than 1
exponent of 10 is a positive whole number
2. value of the exponent
number of places the decimal point must be moved so that the notation is in standard form
3. 0.00563 x 100
For each place the decimal point is moved to the right, subtract 1 from the original exponent
5.63 x 10-3

### Exponential notation: Multiplication

When multiplying numbers written in exponential notation:
1. Multiply digit terms in the normal fashion.
2. Obtain the exponent in the product by adding the exponents of the factors multiplied.
3. If necessary, adjust the exponent to leave just one digit to the left of the decimal point.
(1.25x105) x (4.0x10-2) = (1.25x4.0) x 105+(-2) = 5.0x103

### Exponential notation: Division

When dividing numbers written in exponential notation:
1. Divide the digit terms in the normal fashion.
2. Obtain the exponent in the quotient by subtracting the exponent of the divisor from the exponent of the dividend.
3. If necessary, adjust the exponent to leave just one digit to the left of the decimal point.
(7.5x106) / (3.0x10-2) = (7.5/3.0) x 106-(-2) = 2.5x108
dividend     divisor

## Expressing the Uncertainty (Reproducibility) in Measured Quantities Using Significant Figures

14.62 mL: implied precision +/- 0.01 mL
In this measured quantity, the significant figures are those digits known precisely (namely 1, 4, and 6; these digits are known with a high degree of confidence) plus the last digit (2) which is estimated or is approximate

### Guidelines for counting significant figures

1. Numbers Always Considered Significant
1. all non-zero digits
2. zeros between non-zero digits
3. in numbers containing a decimal point, all zeros written to the right of the rightmost non-zero digit
300.16, 1.0200, and 1,000.0 all contain 5 significant figures.
2. Numbers that are NEVER Significant
Zeros written to the left of the leftmost non-zero digit (these merely indicate the placement of the decimal point)
0.00416 and 0.00000100 both contain three significant figures
3. Trailing Zeros in Numbers Containing No Decimal Point
• Zeros trailing to the right of the rightmost non-zero digit may or may not be significant
For example, the number 100 may have one sig. fig. (100), two sig. figs. (100), or three sig. figs. (100)
• Remove ambiguity by expressing the number using scientific notation
100 expressed as:
• 1 sig. fig. (1x102)
• 2 sig. fig. (1.0x102)
• 3 sig. fig. (1.00x102)
4. Exact Numbers
1. Numbers derived from definition or through counting
2. Numbers considered to be "infinitely precise" (not subject to errors in measurement)
Exact numbers have no effect on the precision expressed in a numerical calculation
 12 inches = 1 ft 1 liter is 1,000,000 mL 1 hr. = 3600 sec. 42 students enrolled in a class

### Significant Figures: Multiplication and Division

The result of these operations will contain the same number of significant figures as the quantity in the calculation having the fewest number of significant figures.
 0.942 atm x 23.482 L n = ------------------------------------------- = 0.864826127 mol = 0.865 mol 0.08205 L atm / mol K x 311.73 K round to 3 sig. fig.

### Significant Figures: Addition and Subtraction

The result must be expressed with the same number of decimal places (i.e., the same absolute uncertainty) as the quantity carrying the least number of decimal places (i.e., the least precisely determined quantity)
 implied precision correct precision 25.6854 g 25.6854 +/- 0.0001 g +0.17     g +0.17     +/- 0.01     g round off 25.8554 g 25.8554 +/- 0.0001 g -- > 25.86 g +/- 0.01 g